Eigenvalues of Robin Laplacians in infinite sectors

For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$, with the Robin boundary condition $partial_ u u=gamma u$, where $partial_ u$ stands for the outer normal derivative and $gamma>0$. The essential spectrum of $T^gamma_alpha$ does not depend on the angle $alpha$ and equals $[-gamma^2,+infty)$, and the discrete spectrum is non-empty iff $alpha<fracpi 2$. In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle $alpha$. In particular, there is just one discrete eigenvalue for $alpha ge frac{pi}{6}$. As $alpha$ approaches $0$, the number of discrete eigenvalues becomes arbitrary large and is minorated by $kappa/alpha$ with a suitable $kappa>0$, and the $n$th eigenvalue $E_n(T^gamma_alpha)$ of $T^gamma_alpha$ behaves as [ E_n(T^gamma_alpha)=-dfrac{gamma^2}{(2n-1)^2 alpha^2}+O(1) ] and admits a full asymptotic expansion in powers of $alpha^2$. The eigenfunctions are exponentially localized near the origin. The results are also applied to $delta$-interactions on star graphs.